Convolution Integral

The convolution integral is the mathematical operation that computes the output of a continuous-time LTI system for any arbitrary input. It is written as y(t) = x(t) * h(t) and is computed using the flip-and-slide method, which is one of the most important techniques in signals and systems analysis.

Convolution is not just a mathematical trick. It has a direct physical meaning: the output at any time t is the sum of contributions from all past values of the input, each weighted by the impulse response. This is how linear systems accumulate and process information over time.

Core Concept Explanation

The derivation of the convolution integral rests on the sifting property of the impulse: any signal x(t) can be written as x(t) = integral of x(tau) delta(t - tau) d(tau). Each delta(t - tau) is a shifted impulse at tau, weighted by x(tau).

Because the system is linear, the output is the superposition of responses to each shifted impulse. Because the system is time-invariant, the response to delta(t - tau) is h(t - tau). Summing (integrating) these contributions gives y(t) = integral of x(tau) h(t - tau) d(tau).

The flip-and-slide method is the graphical procedure for computing this integral: (1) Write h(-tau) by flipping h(tau) about zero. (2) Slide the flipped version by t to get h(t - tau). (3) Multiply x(tau) by h(t - tau). (4) Integrate the product over all tau to get y(t). Repeat for all t to get the full output.

Mathematical Expression

The convolution integral is defined as:

y(t) = x(t) * h(t) = integral[-inf to +inf] x(tau) h(t - tau) d(tau)

An equivalent form is obtained by substitution (commutativity):

y(t) = integral[-inf to +inf] h(tau) x(t - tau) d(tau)

For causal systems (h(t) = 0 for t < 0) and causal inputs (x(t) = 0 for t < 0), the limits simplify to:

y(t) = integral[0 to t] x(tau) h(t - tau) d(tau)

Practical Understanding

In circuit analysis, convolution computes the output voltage of an RC or RLC circuit for any arbitrary input waveform without re-solving the differential equation. In communications engineering, convolution models how a channel distorts a transmitted signal, as the received signal is the convolution of the transmitted signal with the channel impulse response.

In signal processing, FIR filters (finite impulse response) directly implement convolution. The filter output is computed by sliding the filter coefficients (which are the sampled h[n]) over the input sequence and computing dot products. Understanding convolution is therefore essential for digital filter design.

Numerical Example

Compute the convolution of x(t) = u(t) (unit step) and h(t) = e^(-2t) u(t) (first-order system impulse response).

Given:
x(t) = u(t)   [unit step, value = 1 for t >= 0]
h(t) = e^(-2t) u(t)   [time constant = 0.5 s]

Why this formula applies:
Both x(t) and h(t) are causal, so lower limit = 0, upper limit = t.

Formula:
y(t) = integral[0 to t] x(tau) h(t - tau) d(tau)
     = integral[0 to t] 1 * e^(-2(t-tau)) d(tau)   for t >= 0

Substitution:
y(t) = e^(-2t) integral[0 to t] e^(2*tau) d(tau)
     = e^(-2t) * [e^(2*tau)/2] from 0 to t

Calculation:
y(t) = e^(-2t) * (e^(2t) - 1) / 2
     = (1 - e^(-2t)) / 2

Final Answer with units:
y(t) = (1/2)(1 - e^(-2t)) u(t)   [dimensionless output, t in seconds]

Exam Tip: For GATE, when both x(t) and h(t) are causal exponentials, use the formula: u(t) * e^(-at) u(t) = (1/a)(1 - e^(-at)) u(t). Forgetting the 1/a factor is the most common error.

Quick Revision

• y(t) = x(t) * h(t) = integral x(tau) h(t-tau) d(tau).
• Three steps: flip h, slide by t, multiply with x, integrate.
• Convolution is commutative: x(t)*h(t) = h(t)*x(t).
• For causal signals, limits change from [0, t].
• u(t) * e^(-at)u(t) = (1/a)(1 - e^(-at)) u(t).
• In the Laplace domain, convolution becomes multiplication: Y(s) = X(s) H(s).
• GATE trap: Limits of integration change based on support of x(tau) and h(t-tau). Always check overlap region carefully.